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Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^q(\Omega), \ \ q>n/2.$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})$?

Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^q(\Omega), \ \ q>n/2.$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})\cap W^{2,1}{(B)}$?

Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^p(\Omega).$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})$?

Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^q(\Omega), \ \ q>n/2.$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})$?

Assume that $u: B\subset R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^p(\Omega).$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})$?

Assume that $u: B\subset\Bbb R^n\to R$ is two times differentiable almost everywhere in the unit ball $B=B^n$ that is continuous on boundary and that solves the PDE almost everywhere $$L u(x):=\sum_{i,j} a_{ij}(x) \partial_{ij}u(x)=f\in L^p(\Omega).$$ Here $L$ is a strongly elliptic operator. Assume in addition that $v\in W^{2,p}({B})$ for some $p>1$ is weak solution of the same differential equation $Lv=f$ having the same boundary value as $u$. Whether $v=u$ if $u\in W^{1,n}({B})$?